Barber J.R. Intermediate Mechanics of Materials

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Problems 555

and c is a constant. Thus, the dimension a is constant along the beam, but b varies

linearly along the beam.

Use the Rayleigh-Ritz method to estimate the critical value of P at which the

beam will buckle. Use a simple parabolic approximation to the buckled shape.

Notice that the weak axis of bending is parallel to side a near the free end, but

parallel to side b near the built-in end. About which axis does your analysis predict

that buckling will occur?

Figure P12.30

12.31. Figure P12.31 shows a beam of ﬂexural rigidity EI and length L, which is

built in at one end and simply supported at the other. It is subjected to a uniformly

distributed transverse load of w

per unit length and a compressive axial force P.

Use the Rayleigh-Ritz method to estimate the maximum lateral displacement

and sketch the result as a function of P. Use a one degree of freedom polynomial

approximation to the deformed shape.

Figure P12.31

w per unit length

556 12 Elastic Stability

12.32. Figure P12.32 shows a mechanism consisting of two identical rigid bars, each

of length L, loaded by a force P. A spring of stiffness k is attached to the pivot point

and is unstretched when the bars are both horizontal. Find the critical force at which

the horizontal conﬁguration becomes unstable.

Section 12.8

12.33–12.38. Figures P12.33–P12.38 show beams loaded in compression with a va-

riety of different supports. In each case, sketch the probable buckled shape at the

ﬁrst buckling force, count the number of quarter waves in the length L, and hence

estimate the buckling force.

Figure P12.32

Figure P12.33

Figure P12.34

Problems 557

Figure P12.35

Figure P12.38

Figure P12.37

Figure P12.36

The Finite Element Method

Most of this book has been concerned with exact analytical methods for solving

problems in mechanics of materials. Real engineering applications seldom involve

geometries or loading conditions exactly equivalent to those analyzed, so the use of

these methods usually involves some approximation. Nonetheless, the advantages to

design offered by a general (symbolic) analytical solution makes them very useful

for estimation, even when the idealization involved is somewhat forced.

Numerical methods are appropriate when no plausible idealization of the real

problem can be analyzed, or when we require results of greater accuracy than the

idealization is expected to produce. By far the most versatile and widely used numer-

ical method is the ﬁnite element method and every engineer arguably needs to have

at least some acquaintance with it to be considered ‘scientiﬁcally literate’. There is

no room in a book of this length to develop the method to the level where the reader

could either write his/her own ﬁnite element code or use a commercial code. Fortu-

nately, most commercial codes are these days sufﬁciently user-friendly that one can

learn to use them with minimal introduction from the manual, supplemented by the

program help menu. All we seek to do here is to explain the fundamental reasoning

underlying the method, introduce some of the terminology, and generally to remove

the ‘mystique’ that can be a barrier to those not familiar with the method.

In the ﬁnite element method, the body under consideration is divided into a num-

ber of small elements of simple shape, within each of which the stress and displace-

ment ﬁelds are represented by simple approximations, usually low order polynomial

functions of position. For example, in the simplest case, the stress in each element

will be taken as uniform.

Historically, the method was ﬁrst developed as an extension of the ‘stiffness ma-

trix method’ of structural analysis, discussed in §3.9. Each element is treated as a

simple structural component, whose elastic stiffness matrix can be determined by

elementary methods. The problem is thereby reduced to the loading of a set of inter-

connected elastic elements. The global stiffness matrix for the problem is assembled

from that for the individual elements as discussed in §3.9.2.

Early researchers used more or less ad hoc methods for determining the element

stiffness matrices, but the ﬁnite element method was placed on a more rigorous foot-

J.R. Barber, Intermediate Mechanics of Materials, Solid Mechanics and Its Applications 175,

2nd ed., DOI 10.1007/978-94-007-0295-0, © Springer Science+Business Media B.V. 2011

560 A The Finite Element Method

ing by the adoption of the principle of stationary potential energy for this purpose.

In this formulation, the ﬁnite element method is exactly equivalent to the use of the

Rayleigh-Ritz method of §3.6 in combination with a piecewise polynomial approxi-

mation function for the deformation of the structure.

More recently, ‘mathematical’ formulations of the ﬁnite element method have

been developed in which the unknown constants deﬁning the piecewise polynomial

approximation (the degrees of freedom) are chosen so as to satisfy the governing

equations (e.g. the equilibrium equation) in a ‘least squares ﬁt’ sense. We already

remarked in §3.5 that the principle of stationary potential energy is an alternative

statement of the equations of equilibrium, so it should come as no surprise that this

formulation leads to the same ﬁnal matrix equation. However, it has certain advan-

tages, notably that it is readily extended to applications outside mechanics of ma-

terials (e.g. heat conduction, electrical conduction or ﬂuid mechanics) and it also

provides additional insight into the mathematics underlying the method.

In this brief introduction to the ﬁnite element method, we shall illustrate and

compare these alternative formulations in the context of the simplest problem in

mechanics of materials — the axial loading of an elastic bar — and then extend the

argument to the bending of beams. However, we ﬁrst need to discuss some aspects

of approximations in general.

A.1 Approximation

All numerical methods require that the functions describing physical quantities such

as stress and displacement be approximated, typically by the sum of a series of sim-

pler functions with initially unknown linear multipliers. For example, we may choose

to approximate the function f (x) by the series

f (x) ≈ f

∗

(x) ≡

∑

i=1

(x) , (A.1)

where C

are the multiplying constants and the v

(x) are known as shape functions,

which must be linearly independent if the approximation is to be well-deﬁned.

A.1.1 The ‘best’ approximation

If we use f

∗

(x) in place of f (x), it is clear that the governing equations [e.g. the

beam bending equation (1.17) or the equation of motion (10.18)] and/or the bound-

ary conditions will not be exactly satisﬁed and we have to choose the constants C

so as to achieve in some sense the ‘best’ approximation. There is no unique deﬁni-

tion of ‘best’ in this context and different criteria might be appropriate in different

applications.

In the Rayleigh-Ritz method (§3.6), we made this choice so as to achieve the

minimum total potential energy

. This has the advantage of reﬂecting features of

the physics of the system, but purely mathematical criteria can be adopted. One such

A.1 Approximation 561

approach is the collocation method, in which the constants C

are chosen so as to

make the approximation exact at a set of N collocation points x

. In other words, the

are determined from the N simultaneous linear algebraic equations

∗

) =

∑

i=1

) = f (x

) ; j = (1,N) . (A.2)

The disadvantage of this method is that the approximation may be very inaccurate

between the collocation points. This is particularly true if power series terms are used

for the shape functions, v

(x), in which case the resulting approximating function

may snake about the exact line, as shown in Figure A.1.

collocation

points

f(x)

f (x)

f, f

Figure A.1: Pathological behaviour that can result from high order power series

approximations using the collocation method

A more stable and reliable method is to choose the constants C

so as to satisfy

the N linear equations

[ f (x) − f

∗

(x)]w

(x)dx = 0 ; j = (1, N) , (A.3)

where the approximation is required in the range a< x < b and the w

(x) are a set of

N linearly-independent weight functions.

A.1.2 Cho ic e of weight functions

Many readers will be familiar with the procedure for ﬁnding the best straight line

through a set of experimental points (x

), k = (1,M) by using a ‘least squares ﬁt’

— i.e. by minimizing the sum of the squares of the error.

562 A The Finite Element Method

This implies minimizing the function

E ≡

∑

k=1

−y

∗

)]

, (A.4)

where

∗

(x) = Ax + B (A.5)

is the equation of the approximating straight line.

This idea can be extended to the approximation of a continuous function f (x) by

minimizing the error deﬁned in the integral form

E ≡

[ f (x) − f

∗

(x)]

dx . (A.6)

This is essentially equivalent to extending the points x

in equation (A.4) to include

all real points in the range a<x<b. If we represent the approximating function f

∗

(x)

in the form of equation (A.1), we have

E =

f (x) −

∑

i=1

(x)

dx (A.7)

and we can minimize E with respect to each of the degrees of freedom C

by enforc-

ing the condition

∂

= 0 ; i = (1,N) . (A.8)

This leads to the system of equations

−2

f (x) −

∑

i=1

(x)

(x)dx = 0 ; j = (1, N) (A.9)

and hence

[ f (x) − f

∗

(x)]v

(x)dx = 0 ; j = (1,N) . (A.10)

Comparison with equation (A.3) shows that this is equivalent to the use of the same

functions v

(x) as both shape functions and weight functions. This is known as the

Galerkin formulation. The problem of ﬁtting a best straight line to a continuous func-

tion corresponds to the special case N =2, v

(x)= 1, v

(x)= x.

The set of simultaneous equations (A.9) can be rewritten in the form

∑

i=1

(x)v

(x)dx =

f (x)v

(x)dx ; j = (1,N) ,

or in matrix form

KC = F , (A.11)

A.1 Approximation 563

where

(x)v

(x)dx (A.12)

f (x)v

(x)dx . (A.13)

Notice that the coefﬁcient matrix K is simply the integral over the domain of the

product of the shape and weight functions.

A.1.3 Piecewise approximat ions

Everything we have said so far could be applied to any form of approximating func-

tion, including for example power series or Fourier series. The distinguishing fea-

ture of the ﬁnite element method is that a piecewise approximation is used. In other

words, the range of the function is divided into a number of elements within each of

which a fairly simple approximate form is used — usually a low order power series.

a b

Figure A.2: A piecewise linear approximating function

The simplest approximation of this kind is the piecewise linear approximation

illustrated in Figure A.2. Here, the range a< x <b has been divided into N elements,

a≡x

<x <x

, x

<x <x

, ..., x

<x <x

N+1

≡b, in each of which the function f (x)

is approximated by a straight line. The (N +1) points x

,..., x

N+1

are known as

nodes and the corresponding values

∗

≡ f

∗

) , i = (1,N + 1) (A.14)

of the piecewise linear approximating function f

∗

(x) are nodal values. The function

∗

(x) is completely deﬁned if all the nodal values are known.

564 A The Finite Element Method

(a)

v (x)

i-1

i+1

(b)

N+1

v (x)

Figure A.3: Shape functions for the piecewise linear approximation, (a) interior ele-

ments, (b) terminal elements

The piecewise linear function of Figure A.2 can be expressed in the form of

equation (A.1) by using shape functions of the form shown in Figure A.3 (a) and

deﬁned by

(x) = 0 ; x < x

i−1

and x > x

i+1

x −x

i−1

−x

i−1

; x

i−1

< x < x

(A.15)

i+1

−x

i+1

−x

; x

< x < x

i+1

If the elements are all of equal length

∆

b −a

these equations simplify to